Concluding that sine and cosine are $2\pi$ periodic from definitions Let $(f,g)$ be a pair of real-valued $C^1$ functions on $\mathbb{R}$ satisfying
$$\forall x\in\mathbb{R}\left(f'(x)=g(x)\quad\text{and}\quad g'(x)=-f(x)\right)$$
$$f(0)=0\quad\text{and}\quad g(0)=1$$
Then it is pretty immediate that $f$ and $g$ are both $C^\infty$ on $\mathbb{R}$, this determines a power series which determines uniqueness of the pair. A cute argument (without power series) shows that these functions  satisfy $f(x)^2+g(x)^2=1$.
Here's my question: how do we deduce that $f(x)$ and $g(x)$ are $2\pi$ periodic? I understand that we could have started the entire axiomatic system by defining $f(x)$ and $g(x)$ in terms of triangles, but since we already have uniqueness from this setup I wonder how we deduce periodicity from here.
 A: Here is a tedious argument that is less slick than Rudin's but perhaps has more geometric content:
The essence of the argument is to show there is an arbitrarily small rotation that is periodic.
Define $c_0 = 0, s_0 = 1, c_{n+1} = \sqrt{{ 1 + c_n \over 2}}, s_{n+1} = \sqrt{{ 1 - c_n \over 2}}$, $Q_n = \begin{bmatrix} c_n & - s_n \\ s_n & c_n \end{bmatrix}$. Note that
$c_n, s_n \ge 0$ and $c_n^2+s_n^2 = 1$. Also, $c_n \uparrow 1$ (and hence $s_n \downarrow 0$).
($Q_0$ is a $90^\circ$ rotation, and $Q_{n+1}$ is a rotation through half of the angle of $Q_n$.)
A little work shows that $Q_n Q_n^T = I$, $Q_{n+1}^2 = Q_n$ and $Q_n \to I$. Furthermore, $Q_n^{4n} = I$ and $Q_0 Q_n^T = Q_n^T Q_0$.
Let $J= Q_0$, and $x = (g,f)^T$. Then $x' = Jx$ and $x(0) = e_1$.
Note that $x'(0) = e_2$, hence there is some $T>0$ such that $x(t) > 0$ (coordinate wise) for all $t \in (0,T]$. In particular, there is some $n$ and some $t^* \in (0,T]$ such that
$x(t^*) = Q_n x(0) = Q_n e_1$.
Now consider $y(t) = Q_n^T x(t+t^*)$, note that $y(0) = x(0)$ and $y'(t) = Q_n^T J x(t+t^*) = J Q_n^T x(t+t^*)= J y(t)$ and by uniqueness we have $x(t+t^*) = Q_n x(t)$.
In particular, $x(kt^*) = Q_n^kx(0)$ and so $x(4nt) = x(0)$. Hence $x$ is periodic.
A: $t \mapsto (g(t), f(t))$ is a movement with speed $1$ on the curve $x^2+y^2=1$. So the movement is periodic with period the length of the curve. 
You can cook up an example of a differential equation that gives a movement with unit speed on, say, an ellipse.  
In general, consider a system of the form $g'(t) = A(g(t), f(t))$, $f'(t) = B(g(t), f(t))$, 
where $(A(x,y), B(x,y))= \lambda(x,y) (-\frac{\partial F}{\partial y}, \frac{\partial F}{\partial x})$. Then $F$ is constant on any trajectory of the system. Assume moreover that the initial point lies on a compact connected component of a level curve of $F$ and $\lambda \cdot \nabla F \ne 0$ on it (so it lies on a closed curve). Then again the solution of the system will be periodic.
In the case of a vector $v$ field tangent to a closed curve $C$, the equation $\frac{d\bf{x}}{dt} = v(\bf{x})$ starting on the curve will have a period 
$$\tau = \int_C \frac{ds}{\|v\|}$$ that can be in principle calculated (approximated). 
